We say that a function $f:U \subset\mathbb{R} \to \mathbb{R}$ (with $U$ open) is differentiable at $x\in U$ if the limit $\lim_{ h \to 0 } \frac{f(x+h) -f(x)}{h}$ exists. This limit is denoted $f'(x)$. This allows a nice linear approximation of $f$ near the point $x \in U$, that is, $f(x+h) = f(x) + f'(x)(h)+\varepsilon(x,h)$ where $\varepsilon(x,h) \to 0$, as $h\to 0$. We generalise the notion of differentiablity to functions $f: U \subset \mathbb{R}^{m}\to \mathbb{R}^{n}$ as the best approximating linear map. # Definitions > [!NOTE] Definition > Let $U \subset \mathbb{R}^n$ be [[Open Sets in Metric Space|open]] and $a\in U$. We say that a [[Function|function]] $f:U \to \mathbb{R}^m$ is **differentiable** at $u$ if there exists a [[Linear Map|linear map]] $L_{a}:\mathbb{R}^n \to \mathbb{R}^m$ such that for all $h\in \mathbb{R}^n$ with $a+h\in U$ we have $f(a+h)=f(a)+L_{a}(h)+R(a,h)$where $\lim_{\substack{ h \to 0 \\ h \neq 0}} \frac{\lVert R(a,h) \rVert}{\lVert h \rVert } = \lim_{ h \to 0 } \frac{\lVert f(a+h) - f(a)-L_{a}(h) \rVert}{\lVert h \rVert} =0. $We use $Df(a)$ to denote $L_{a}\in L(\mathbb{R}^n, \mathbb{R}^m)$, which we call the **derivative** of $f$ at $a$. # Properties ###### Differentiability and continuity Note that [[Differentiablity implies Continuity|differentiability implies continuity]]. However continuity doesn't not imply differentiability (e.g [[Weierstrass function]]). We can use [[Mean Value Theorem (Existence of Point at Which Tangent of Arc is Parallel to Secant Through Endpoints)]] to show that [[Positive Derivative Implies Strictly Increasing Real Function]] & [[Real Function with Zero Derivative is Constant]]. See [[Extrema and Derivatives]]. Note that the [[Derivative of Inverse of Strictly Monotonic Differentiable Real Function|inverse of differentiable univariate real-valued function is also differentiable]]. See [[Derivative of Monomials]]. By [[Linearity of Derivative of Differentiable Vector-Valued Function of Single Real Variable]],